E
extracta mathematicae Vol. 18, N´um. 3, 311 – 314 (2003)
Note on a Banach Space Having Equal Linear Dimension with its Second Dual ´ jtowicz A. Plichko, M. Wo Politechnika Krakowska im. Tadeusza Ko´sciuszki, Instytut Matematyki, ul. Warszawska 24, 31-155 Krak´ ow, Poland Instytut Matematyki, Akademia Bydgoska, pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland e-mail:
[email protected],
[email protected] (Presented by Jes´ us M.F. Castillo)
AMS Subject Class. (2000): 46B03, 46B10
Received July 9, 2003
In the paper [4] it is stated that (E) there exists a Banach space X whose bidual X ∗∗ is isometric to a subspace of X, but not isomorphic to X. Since X is isometric to a subspace of X ∗∗ , (E) gives another solution to a Banach problem, posed in [1], p.193 (and solved for the first time in [2]): Let a Banach space X be isomorphic to a subspace of a Banach space Y and let Y be isomorphic to a subspace of X. Are then X and Y isomorphic? Let us note that this problem becomes essential (and difficult) if one assumes additionally that X and Y are isomorphic to complemented subspaces of each other (the Schroeder-Bernstein problem, solved in the negative by Gowers [5]). The construction in [4] is as follows: one takes Z = Z (0) = 1) and PL1 (0, (2n) Z )`2 . its successive even duals Z ∗∗ , Z (4) , . . ., and X is defined as ( ∞ n=0 ∗∗ Of course, X is isometric to a subspace of X. Next, the author attempts to prove that X and X ∗∗ are non-isomorphic; however, Ezrohi’s arguments at this point are difficult to follow. In the proof he refers to his Ph. D. result which, in our opinion, is incorrect. On the other hand, it is known that L1 is complemented in L∗∗ 1 (see e.g.[8], Proposition 8.3 (v), p. 113; cf. [9]). It ∗∗ follows that L1 ⊕ L1 is isomorphic to L∗∗ 1 , and hence, X is isomorphic to X ∗∗ and the construction is improper. Nevertheless, this space X is a simple example of a nonreflexive Banach lattice isomorphic to its bidual. (Recall, 311
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´ jtowicz a. plichko, m. wo
that the famous James space [6] appears to be a solution to another problem of Banach: whether there is a nonreflexive Banach space which is isomorphic to its bidual?) In this note we show that the assertion (E) holds true for X constructed as above with the basic space c0 (=Z) instead of L1 (0, 1). We recall that a Banach space Z is a Grothendieck space if the weak and weak* convergence of sequences in Z ∗ coincide. A compact Hausdorff space K is called Stonian [quasi-Stonian, resp.] if each open [and Fσ , resp.] ˇ subset of K has an open closure. The Cech-Stone compactification βN of the discrete space N, of all positive integers, is a sample Stonian space. Every C(K)-space, with K quasi-Stonian, is a Grothendieck space; in particular, `∞ = C(βN) is of this type (see [8], pp. 131-132; or [7], pp.111 and 348360). Moreover, from the Kakutani-Krein theorem (see [8]; Corollary 1, p. 104) it follows that both L∞ (µ)-space and the biduals of C(K)-spaces, for K arbitrary compact Hausdorff (being isometric to C(L)-spaces for some L Stonian) are Grothendieck spaces ([8], p. 121). It easy to check that the class of Grothendieck spaces is closed with respect to quotient operations and linear isomorphisms. It is known that separable quotients of Grothendieck spaces are reflexive ([7], Proposition 5.3.2); in particular no complemented subspace of a Grothendieck space is isomorphic to c0 .
(1)
We note that the class of Grothendieck spaces is also closed with respect to `p -sums, 1 < p < ∞, and we apply this property to construct a Banach space X such that X ∗∗ is isometric to a complemented subspace of X, with X and X ∗∗ non-isomorphic. The details follow. Lemma 1. Let (Z Pn ) be a sequence of Grothendieck spaces, and let 1 < p < ∞. Then Z := ( ∞ n=1 Zn )`p is a Grothendieck space. Theorem 1. Put Z0 = c0 , and Zn+1 = (Zn )∗ , n ≥ 0. Then for the space P∞ X := ( n=0 Z2n )`p , 1 < p < ∞, we have: X ∗∗ is isometric to a complemented subspace of X but X and X ∗∗ are non-isomorphic. The proof of Lemma 1. This lemma is rather known (at least if Z1 = Z2 = . . . [3]) and we present the proof for sake of completeness. We apply the following corollary of the Banach-Steinhaus theorem:
space having equal dimension with second dual
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(∗) Let (Tn ) be a uniformly bounded sequence of linear operators defined on a Banach space X and let D be a dense subset of X. If kTn xk → 0 as n → ∞ for every x ∈ D, then kTn xk → 0 for every x ∈ X. P ∗ = ( ∞ Z ∗ ) , with 1/p + 1/q = 1, and hence Z ∗∗ = Notice first that Z n=1 n `q P P∞ ∗∗ ) . Thus every f ∈ Z ∗ is of the form f (z) = ( ∞ Z ` p n=1 n n=1 fn (zn ), where P q < ∞. z = (zn ) ∈ Z and fn ∈ Zn∗ , n ≥ 1, with ∞ kf k n n=1 Let (f (m) ) be a weak*-null sequence in Z ∗ : ∞ X
fn(m) (zn ) → 0,
as m → ∞,
n=1
for every sequence (zn ) ∈ Z with zn ∈ Zn ; in particular, lim f (m) (zn ) m→∞ n
=0
for every n.
(2)
Since Zn ’s are Grothendieck, from (2) we obtain that for every Fn ∈ Zn∗∗ , n = 1, 2, . . ., we have lim Fn (fn(m) ) = 0. (3) m→∞
Now let D denote the dense in Z ∗∗ set of the elements F of the form F = P k ∗∗ n=1 Fn , where Fn ∈ Zn , and k = 1, 2, . . ., and consider the linear functionals ∗∗∗ ∗∗ ∗∗∗ (F ) := F (f (m) ). By (3), these functionals zm , m ≥ 1, on Z of the form zm ∗∗∗ ) is weak*-null in Z ∗∗∗ or, fulfil the assumptions of (*), and therefore (zm (m) ∗ equivalently, (f ) is weak-null in Z . The proof of Theorem 1. We have that Z2n = C(Kn ), where n ≥ 1 with K1 = βN, and the remaining Kn ’s are ”gigantic” Stonian spaces (see the remarks preceding Lemma 1). Therefore, by Lemma 1, X ∗∗ is a Grothendieck space, which evidently is isometric to a complemented subspace of X and which, by property (1), cannot be isomorphic to X. Remark. Of course, the space X in Theorem 1 is not isomorphic to a complemented subspace of X ∗∗ , and hence the following question seems to be quite natural. Let X be isomorphic to a complemented subspace of X ∗∗ and let X ∗∗ be isomorphic to a complemented subspace of X. Are then X and X ∗∗ isomorphic? Professor Galego has remarked that this question is connected with the following one: Let X be complemented in X ∗∗ . Is X ∗∗ isomorphic to X ⊕ X ∗∗ ? In particular, is X ∗∗∗ isomorphic to X ∗ ⊕ X ∗∗∗ ? Evidently, it is true if X is isomorphic to its square.
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References [1] Banach, S., “Th´eorie des Op´erations Lin´eaires”, Monografie Matematyczne, Warszawa-Lw´ow, 1932.
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[3] Bombal, F., Operators on vector sequence spaces, London Math. Soc. Lecture Note Ser., 140 (1989), 94 – 106.
[4] Ezrohi(Ezrokhi), I.A., On linear dimension, Dokl. Akad. Nauk SSSR, 62 (1948), 35 – 38 (Russian).
[5] Gowers, W.T., A solution of the Shroeder-Bernstein problem for Banach spaces, Bull. London Math. Soc., 28 (1996), 297 – 304.
[6] James, R.C., Bases and reflexivity of Banach spaces, Ann. of Math., 52 (1950), 518 – 527.
[7] Meyer-Nieberg, R., “Banach Lattices”, Springer-Verlag, Berlin, 1991. [8] Schaefer, H.H., “Banach Lattices and Positive Operators”, Springer-Verlag, Berlin, 1974.
´ jtowicz, M., Contractive projections onto isometric copies of L1 (ν) in [9] Wo
strictly monotone Banach lattices, Bull. Polish Acad. Sci. Math., 51 (2003), 1 – 12.
